Colorable Hierarchically Hyperbolic Space

• Colorable hierarchically hyperbolic space is a quasi-geodesic metric space with a hierarchical structure and a combinatorial coloring that partitions domains into pairwise transverse classes.
• Its stable cubulation theorem enables finite subsets to be quasi-isometrically embedded in CAT(0) cube complexes, ensuring robust geometric approximation and resistance to perturbations.
• The framework unifies combinatorial, geometric, and algebraic insights, with significant applications to mapping class groups, Teichmüller spaces, and right-angled Artin groups.

A colorable hierarchically hyperbolic space (colorable HHS) is a quasi-geodesic metric space $(X,d)$ equipped with a hierarchically hyperbolic structure that admits a combinatorial partitioning—called coloring—of its domain set, enabling robust cubical modeling for finite configurations. This property facilitates stable quasi-isometric approximations of hulls by CAT(0) cube complexes and underpins refined connections with cubical geometry, coarse medians, and algebraic structure in groups. Colorability, which mandates that domains of the same color are pairwise transverse, has fundamental applications to mapping class groups, Teichmüller spaces, RAAGs, and large-type Artin groups, among others.

1. Formal Definition and Structural Axioms

A hierarchically hyperbolic space $(X,\mathfrak S)$ comprises:

• A quasi-geodesic metric space $(X,d)$.
• An index set $\mathfrak S$ of domains, with each $U \in \mathfrak S$ associated to a $\delta$-hyperbolic space $\mathcal C U$ and a coarse Lipschitz projection $\pi_U: X \to 2^{\mathcal C U}$, $\diam(\pi_U(x)) \le \xi$.
• Relations on $\mathfrak S$: nesting $\sqsubseteq$, orthogonality $\perp$, transversality $\pitchfork$, where for any distinct $U, V$, exactly one holds: $U \sqsubseteq V$, $U \perp V$, $U \pitchfork V$.
• Relative projections $\rho^U_V \subset \mathcal C V$ for $U \sqsubset V$ or $U \pitchfork V$, of bounded diameter.

These satisfy consistency, bounded-geodesic-image, large-link, partial-realization, and uniqueness axioms. The distance formula for $K \gg 0$ gives:

$$d_X(x,y) \asymp \sum_{U \in \mathfrak S} [ d_U(x,y) ]_K,$$

where $d_U(x,y) = \diam(\pi_U(x) \cup \pi_U(y))$.

A colorable HHS is one whose domain set $\mathfrak S$ can be partitioned,

$\mathfrak S = \bigsqcup_{i=1}^N \mathfrak S_i,$

so that each color-class $\mathfrak S_i$ consists of pairwise transverse domains and is permuted by the automorphism group $G < \mathrm{Aut}(\mathfrak S)$ (Durham et al., 2020, Durham, 30 Dec 2025, Durham et al., 23 Apr 2025, Bongiovanni et al., 21 Aug 2025).

2. Stable Cubulation and Hyperplane Stability

A central theorem for colorable HHSs, the stable cubulation theorem, states: For any finite $F \subset X$, $|F| \le k$, there exists a CAT(0) cube complex $\mathcal Y_F$ of dimension $\le \xi$ and a $(K,K)$-quasi-isometric embedding

$\Phi_F: \mathcal Y_F \rightarrow X,$

with $$d_{\mathrm{Haus}}(\Phi_F(\mathcal Y_F), H_\theta(F)) \le K$$, where $H_\theta(F)$ is the hierarchical hull,

$H_\theta(F) = \{ x \in X : \forall U \in \mathfrak S, \pi_U(x) \subset N_\theta(\hull_{\mathcal C U}(\pi_U(F))) \}.$

Moreover, for $F'$ differing from $F$ by small Hausdorff moves, $\mathcal Y_F$ and $\mathcal Y_{F'}$ differ by at most $N$ hyperplane insertions/deletions, allowing for a coarse convex embedding and guaranteeing stability under perturbations (Durham et al., 2020, Durham, 30 Dec 2025, Durham et al., 23 Apr 2025).

This cubulation is constructed by identifying relevant domains with large projections, modeling hulls in each $\mathcal C U$ by trees, and then cubulating the combined hulls using Sageev’s dual-cube-complex procedure. Coloring constrains wall and subtree choices, yielding stable cubical approximations.

3. Barycenters, Bicombings, and Hierarchy Paths

In a $G$-invariant colorable HHS, one canonical barycenter map $\tau: X^k \to X$ can be defined by pushing the barycenter in the cube complex $\mathcal Y_F$ (obtained using normal contractions) back to $X$ via $\Phi_F$. This barycenter is symmetric in its arguments and moves by at most $\kappa$ under small perturbations of its input, ensuring stability:

• $\tau$ is $\kappa$-coarsely Lipschitz and $\kappa$-coarsely equivariant under $\mathrm{Aut}(X,\mathfrak S)$.
• $\tau(x_1, ..., x_k) \in H_\theta(\{x_i\})$.
• If $k$-tuples differ by at most one in each coordinate, barycenters move by at most $\kappa$ (Durham et al., 2020).

For bicombings, pushing canonical cube-complex normal-path bicombings through $\Phi_F$ provides a discrete, bounded, quasi-geodesic bicombing by hierarchy paths. This yields:

• Coarsely equivariant bicombing in Teichmüller spaces with both Teichmüller and WP metrics.
• Mapping class groups $\mathrm{MCG}(\Sigma)$ and Teichmüller spaces are semihyperbolic (Durham et al., 2020, Durham, 30 Dec 2025).

4. Algebraic Characterizations and Coarse Median Equivalences

For groups, a colorable hierarchically hyperbolic group (colorable HHG) admits a coloring such that there is no $G$-orbit containing two orthogonal domains. In the context of free-by-cyclic groups $G = F_n \rtimes_\phi \mathbb Z$, colorability is characterized algebraically by the absence of unbranched blocks, that is, there do not exist three blocks (maximal virtually $F \times \mathbb Z$ subgroups) intersecting in a subgroup that is not virtually cyclic.

For $G = F_n \rtimes_\phi \mathbb Z$, the following equivalence holds:

• $G$ has unbranched blocks.
• $G$ admits a coarse median of finite rank.
• $G$ is virtually colorable HHG.
• $G$ is quasi-isometric to a finite-dimensional CAT(0) cube complex.
• $G$ does not admit a quasi-isometrically embedded 2-dimensional richly-branching flat (Bongiovanni et al., 21 Aug 2025).

A criterion, “excessive linearity,” for CT maps associated to $\phi \in \mathrm{Out}(F_n)$, obstructs colorability. No excessive linearity is equivalent to unbranched blocks and thus colorable HHG structure.

5. Asymptotically CAT(0) Metrics and Z-Structures

Colorable HHGs admit asymptotically CAT(0) metrics, meaning that the CAT(0) inequality holds up to a sublinear error $\kappa(r) = O(\sqrt{r})$ on quasi-geodesic triangles. This is established by constructing a family of metrics $\{d^p\}$ via cubical models for hulls and correcting for triangle errors:

$$d_X(p,q) \le d_{\mathbb E^2}(\bar p, \bar q) + \kappa( \min \{ d(p, \partial \Delta), d(q, \partial \Delta) \} ),$$

where $(X,\mathfrak S)$ is colorable. Asymptotic cones of $(X,d_2)$ are CAT(0) (Durham et al., 23 Apr 2025).

As a consequence, colorable HHGs admit Z-structures, constructed via contractible Vietoris–Rips complexes at scale. The boundary $\partial X$ of sublinearly Morse rays forms a compact ER with $\partial X$ as a Z-set. These structures provide a means to prove the Farrell–Jones conjecture for colorable HHGs under amenable action conditions on subgroups stabilizing non-maximal domains (Durham et al., 23 Apr 2025).

6. Key Examples and Applications

Principal examples of colorable HHSs and HHGs include:

• Mapping Class Groups ($\mathrm{MCG}(S)$), with domains colored via subsurface combinatorics (Bestvina–Bromberg–Fujiwara's chromatic method).
• Teichmüller space and its WP metric variant, with domains being isotopy classes of non-annular subsurfaces.
• Right-angled Artin groups (RAAGs), with join-subgraphs of defining graph being the domains and coloring corresponding to the chromatic number of the complement graph.
• Cubical groups, certain 3-manifold groups, extra large-type Artin groups (Durham et al., 2020, Durham, 30 Dec 2025, Durham et al., 23 Apr 2025).

These groups are colorable, possess stable cubulations, coarse barycenters, semihyperbolicity, asymptotically CAT(0) metrics, and satisfy the Farrell–Jones conjecture. Known non-colorable examples are rare and require careful engineering (Durham et al., 23 Apr 2025).

7. Significance and Unifying Framework

Colorability in hierarchically hyperbolic spaces and groups unifies combinatorial, geometric, and algebraic control, allowing robust finite CAT(0) modeling for hulls, stable bicombings, barycenters, and sublinear deviation from CAT(0) geometry. It is the core property distinguishing “naturally occurring” HHGs (mapping class groups, RAAGs) from pathological cases, and enables the transfer of cubical techniques and contractibility properties across geometric group theory, geometric topology, and coarse geometry.

A plausible implication is that colorability, by controlling the interaction of domains and ensuring stability of cubical models under perturbation, is instrumental for developing extensions to metric invariants and actions on cube complexes, facilitating results for large-type Artin groups and beyond.